Drag reduction is one possible measure of the performance of surfactant-contaminated SHSs.
An alternative measure, commonly used for laminar flows over SHSs, is the effective slip length \(\lambda _e\)  [1]}, [2]}, defined as the uniform slip length applied to the top and bottom boundaries of an equivalent channel of the same height as the SHS channel and which has the same flow rate as the SHS channel under the same pressure gradient [3]}.
To evaluate \(\lambda _e\) , we integrate the leading-order streamwise momentum equation ( REFb) for an equivalent channel with the mixed boundary conditions (REF , REF ) replaced by \(\lambda _e u_{0y} - u_0 = 0\)  on \(\mathcal {I}\) , \(\mathcal {R}\)  and \(\mathcal {S}\) .
We obtain \(u_0 = \check{U} \Delta p_{0x}\)  where \(\check{U} = y(y-2)/2 - \lambda _e\)  and \(p_{0x}\)  is the same pressure gradient as in the SHS channel.
The flux is \(\check{Q} = \int _{z = - P_z}^{P_z} \int _{y = 0}^{2} \check{U} p_{0x} \text{d}y \text{d}z = (\breve{Q} - 2 P_z \lambda _e)p_{0x}\) , or by integrating over one period \(\check{Q} = (\breve{Q} - 2 P_z \lambda _e) \Delta p_{0} / 2\) .
Equating the flux of the equivalent channel with the flux of the SHS channel, \(\tilde{Q}p_{0x} + \bar{Q}\Gamma _{0x} = 1\) , we find
\( \lambda _e = \frac{{DR}_0(\Delta p_R - \Delta p_U)}{P_z \Delta p_R(\Delta p_U {DR}_0 + \Delta p_R (1-{DR}_0))},\) 
